Vector-Valued Wavelet Bases as Hilbert $\mathbb{M}_m(\mathbb{R})$-Module Bases: A Construction from Scalar Wavelets

What the math gives to ML

The paper provides a constructive way to turn scalar tensor-product wavelets into vector-valued bases whose basis elements mix signal components rather than processing channels independently. The transferable asset is a fixed, perfectly reconstructing multiscale analysis bank with compact support, vanishing moments, regularity, and structured cross-channel coupling. A practical neural-network use is a wavelet front end or downsampling block in which the cross-paired vector wavelets replace separate depthwise wavelet transforms, followed by lightweight learned channel mixing while retaining an exact inverse and a Parseval-style energy diagnostic.